ECE 333 GREEN ELECTRIC ENERGY. 6. Limits on Conversion of Wind Into Electricity

Transcription

1 ECE 333 GREEN ELECTRIC ENERGY 6. Limits on Conversion of Wind Into Electricity George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana Champaign ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1

2 CONVERSION OF WIND INTO ELECTRICITY q We analytically characterize the power in wind as a cubic function of the wind speed v q The wind energy is in the form of kinetic energy, which is extracted from the wind and used to rotate the generator shaft mounted in the nacelle q We examine the constraint the so called Betz limit that restricts the ability of a wind turbine to convert kinetic energy into mechanical energy to spin the turbine generator shaft ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 2

3 THE BETZ LIMIT q The limit was derived in 1919 by Albert Betz, a German physicist q We consider the wind as it passes through a wind turbine and we examine the wind stream q We explain the conceptual basis on the limit of the conversion of wind into electricity using the diagram below ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 3

4 THE BETZ LIMIT upwind v v r downwind v d rotor area a stream tube is formed by the air mass passing through the turbine ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 4

5 q Clearly, the turbine cannot extract all the kinetic energy in the wind because that implies that the air would have to stop completely after passing the turbine an impossible situation since it would prevent all the continuing wind to pass through the rotor q Furthermore, the downwind velocity equal v THE BETZ LIMIT since that would imply that no energy is extracted by the turbine v d cannot ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 5

6 THE BETZ LIMIT q Betz formulated the relationship to determine the maximum mechanical power obtainable from wind q We focus on what happens with the wind as it passes through the plane of the rotor blades at velocity, with where, we explicitly take into account that as the wind mass of air goes through the stream tube, the downwind speed is lower than the upwind speed v r v d < v r < v ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 6,

7 THE BETZ LIMIT q The conservation of energy implies that kinetic energy upwind = kinetic energy downwind + energy extracted by blade rotor q Therefore, as the mass flow rate throughout the stream tube remains unchanged, the power extracted by the rotor blades is p r kinetic dm dt d = energy energy dt = 1 2 dm dt upwind ( v 2 2 v ) d kinetic downwind ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 7

8 q Now, we can determine anywhere in the stream tube and at the rotor blade plane since and we assume that so that THE BETZ LIMIT dm dt dm dt = ρ avr v r = v + v d 2 p r = 1 2 ρ a v + v d 2 v 2 2 v d ( ) ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 8

9 THE BETZ LIMIT q We introduce the ratio λ defined by v d = λ v so that the expression for p r becomes p r = 1 2 ρ av 1 + λ 2 v 2 1 λ 2 ( ) = 1 ρav λ 1 λ 2 2 ( )( ) 3 2 power in the wind fraction extracted ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 9

10 THE BETZ LIMIT q We can think of the fraction of power extracted by the rotor as the efficiency η r of the rotor η r = 1 ( λ )( 1 λ 2 ) so that p r = 1 2 ρ av 3 η r ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 10

11 THE BETZ LIMIT q To determine the maximum rotor efficiency we evaluate the derivative of p r w.r.t. λ dp r dλ = 1 2 ρ av λ ( ) ( )( 2λ ) + ( 1) 1 λ 2 = 1 4 ρ av 3 2λ 2 2λ + 1 λ 2 = 1 4 ρ av 3 1 2λ 3λ 2 = 1 4 ρ av λ ( )( 1 3λ ) ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 11

12 THE BETZ LIMIT q We set dp r dλ to be 0, and we solve for λ q The only physically meaningful solution is λ = 1 3, i.e., the efficiency is maximized when the ratio of v d to v is 1/3 so that ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 12

13 THE BETZ LIMIT η r = = = 59.3 % q This optimal theoretical efficiency better known as the Betz efficiency cannot be higher than 59.3 %; this value is the essence of the Betz limit ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 13

14 THE BETZ LIMIT q The Betz limit implies that even under ideal conditions less than 60 % of the power in wind can be extracted and in actual systems, the best that is attainable is, typically, below 50 %; in other words, at most half of the energy in wind can be converted into mechanical energy to rotate the generator shaft ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 14

15 TIP SPEED RATIO q The tip speed of the rotor is a function of the rate of rotation of the rotor specified by its r.p.m.: in each revolution of the rotor, the tip traverses a distance π d and so the tip speed is ( π d)( r. pm..) q A convenient way to express rotor efficiency is in terms of the tip speed ratio τ, where τ = rotor tip speed v = (r.p.m.) i min 60 sec i π d v ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 15

16 TIP SPEED RATIO q Studies indicate that modern turbines attain maximum efficiency for 4 τ 6 : the tip of the blade moves 4 6 times faster than the wind speed q It follows that for maximum efficiency it is desirable that turbine blades change their speed as wind speed changes as is the case in variable speed generators ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 16

17 INTERPRETATION OF η r q Betz s law states that the maximum power that we can extract from wind is p r = 1 2 ρ av q Engineers define the efficiency as the ratio of the output to the input quantity p out η r = p in and so the natural question is what is p in ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 17

18 INTERPRETATION OF η r q A convenient way to think about is that is the power in the wind prior to the installation of a turbine: absent the turbine, pin = p in and so q The Betz efficiency determines the limit on the v 1 ρ av 2 r 3 = v p in conversion of p in into mechanical power to rotate the generator shaft ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 18

19 WIND TURBINE GENERATORS q The wind turbine generators or wind energy conversion systems may be classified into two principal categories m variable speed rotors m fixed speed rotors q The variable speed turbines have the ability to take advantage of the fact that wind speed varies ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 19

20 WIND TURBINE GENERATORS and adjust the rotor speed so as to optimally match wind speed q The fixed speed rotor generators are simpler but cannot operate at optimal efficiency; moreover, the stresses from the rapidly varying wind speeds, typically, require sturdier design of such turbines ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 20

21 WECS TYPE A q The turbine design uses an induction generator connected to a fixed speed wind turbine q This design needs two additional components for grid connection: ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 21

22 WECS TYPE A m a soft starter to decrease current transients during startup phase m a capacitor bank to compensate for reactive power q As a result of the capacitor bank, the generator can work close to a zero value generation or consumption of reactive power q Unfortunately, such capacitive compensation does not provide flexible reactive power control by the wind turbine ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 22

23 WECS TYPE B q The Vestas developed type B WECS generator is designed to work with a limited variable speed ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 23

24 WECS TYPE B wind turbine q The turbine uses the variable resistor in the rotor to control the real power output q The capacitor bank and soft starter device roles are analogous to those in the the type A design ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 24

25 WECS TYPE C q This design uses two AC/DC converters rated at 25 % of total generator power with a capacitor between them to control the WECS q The wound rotor induction generator topology is also known as a doubly fed induction generator (DFIG) ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 25

26 WECS TYPE C q The term doubly comes from the fact that the rotor winding is not short circuited as in the classical singly fed induction machine but a voltage is induced from the rotor side converter q Depending on the operating scheme, they can keep a constant value of reactive power or keep the terminal voltage constant q The WECS type C is the most widespread of all wind turbines on the market ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 26

27 WECS TYPE D q The type D design uses a full scale frequency converter with different types of generators q The most common implementation is the permanent magnet synchronous generator (PMSG) ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 27

28 WECS TYPE D q This design allows full control over active and reactive power production and has a high wind energy extraction value q Full power control improves power and frequency stability in the grid and reduces the short circuit power q Most type D designs do not need a gearbox a key advantage of type D WECS ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 28

29 WIND TURBINE CLASSIFICATION wind energy conversion systems variable-speed turbines indirect drive with gearbox direct drive without gearbox fixed-speed turbines wound-rotor synchronous generator permanent-magnet synchronous generator squirrel-cage induction generator doubly-fed induction generator wound-rotor induction generator with variable R wound-rotor synchronous generator permanent-magnet synchronous generator squirrel-cage induction generator ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 29